Distribution problems for L-functions and number fields

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This research program is in the field of number theory, which is the part of mathematics that deals with properties of integers and solutions of equations with integer coefficients. Such equations are called Diophantine equations. Elliptic curves are one of the fundamental objects of study in the theory of Diophantine equations. Many number theoretical problems can be studied more generally in certain subsets of complex numbers called number fields. Over the past decade, I have studied several number theoretical problems related to elliptic curves and number fields from an analytic point of view and by employing L-functions which are fundamental tools in analytic number theory. I plan to further my research in such problems over the next five years. I aim to extend some of the fundamental results of analytic number theory, as originally stated for the Riemann zeta function and Dirichlet L-functions, to the general setting of automorphic L-functions; to prove new value distribution theorems for certain families of L-functions and explore the applications of such results in the distribution of invariants of number fields such as Euler-Kronecker constants; and to investigate distribution problems arising in geometric settings formulated in terms of division fields of elliptic curves and their associated L-functions. The outcome of this research will enrich our knowledge in several key areas of number theory and will help to pave the way for further studies in the field. In addition, this research program provides excellent training opportunities for students and postdoctoral fellows.